I am reading the following article, https://arxiv.org/abs/1812.08767, and I am trying to go though the derivation of McLachlan's variational principle for imaginary time evolution. The derivation is done in section B.1.2.

I am confused about going from this equation:

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to this equation (assuming real $\theta_i$):

enter image description here

I think the $x$ at the end is a typo. What I dont understand is how the variation $\delta$ removes the need to sum over $i$, going from eq. (120) to (123). They also remove the $\dot{\theta}_i$ factor and seem to replace it with $\delta \theta_i$. I also dont understand why the final term in eq. (120) disappears.

I think I am overall a little confused as to what the variation $\delta$ actually does to an equation. Could anyone explain?

  • $\begingroup$ I just learned that that the L2 norm (120) is differentiated with respect to $\dot{\theta}_i$. $\endgroup$
    – QCQCQC
    Nov 15 '20 at 14:53
  • $\begingroup$ This leaves one final confusion. How can one see that the normalization of the wavefunction introduces the $E_\tau$ term in the minimization of the L2 norm? $\endgroup$
    – QCQCQC
    Nov 15 '20 at 14:58

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